Further Mathematics · AQA 7367 · Option S · Topic SD

Continuous random variables (CRVs)

Continuous random variables: probability density functions, cumulative distribution functions, means, variances and the rectangular distribution.

8 sub-topics
6 in Year 12, 2 in Year 13
86 practice questions
8 animated lessons

Practice questions (86) Lessons Worksheets All AQA Further Mathematics topics

What the specification says

Every sub-topic AQA can examine in this topic, with its reference number from the official specification. The Y12 and Y13 tags show when each sub-topic is usually taught.

SD1Probability density functionsY12
SD2Probability over an intervalY12
SD3Median and quartilesY12
SD4Mean, variance and standard deviationY12
SD5Expectation and variance of functions of a CRVY12
SD6Cumulative distribution functionsY13
SD7The rectangular distribution, its mean and varianceY13
SD8Expectation and variance of sums of independent random variablesY12

Try 3 questions

Easy, medium and hard questions from this topic, each with a worked solutions and the mark for every step. All 86 questions in this topic, free, no account needed.

Continuous random variables (CRVs)

Easy

Median and quartiles from the cdf · Year 12 · 1 mark

$X$ is a continuous random variable with cumulative distribution function $F$ . Which equation gives its median?
(1)

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Solve $F(x) = p$ for the value below which a proportion $p$ of the distribution lies. $C$ : $F(x) = 0.5$

Total1

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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Continuous probability distributions

Not needed for Year 12Medium

Continuous random variables · Year 13 · 5 marks

The continuous random variable

X

has probability density function

f(x) = kx(2 - x)

for

0 \leq x \leq 2

, and 0 otherwise.

(a)
Show that $k = \frac{3}{4}$ .
(2)
(b)
Find $\mathrm{P}(X < \frac{2}{3})$ .
(2)
(c)
Write down $\mathrm{E}(X)$ , giving a reason.
(1)

Check your answer before you look at the solution

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Part (a)2 marks

The total probability is 1. $\int_0^{2} kx(2 - x)\,\mathrm{d}x$ ${}= 1$
$k \times \frac{4}{3} = 1$

Part (b)2 marks

$\int_0^{\frac{2}{3}} \frac{3 x \left(2 - x\right)}{4}\,\mathrm{d}x$
$= \frac{7}{27}$

Part (c)1 mark

The pdf is symmetric about its midpoint. $1$

Total5

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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Continuous probability distributions

Not needed for Year 12Hard

The cumulative distribution function · Year 13 · 6 marks

The continuous random variable

X

has cumulative distribution function

F(x)

{}= 0 \ (x < 0), \ F(x)

{}= \dfrac{x^2}{4} \ (0 \leq x \leq 2), \ F(x)

{}= 1 \ (x > 2)

.

(a)
Find the probability density function of $X$ .
(2)
(b)
Find the exact median and upper quartile of $X$ .
(3)
(c)
Find $\mathrm{P}(X > 1)$ .
(1)

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Part (a)2 marks

$f(x) = F'(x) = \frac{x}{2}$
$0 \leq x \leq 2; \ 0 \text{ otherwise}$

Part (b)3 marks

$F(m)$ ${}= \tfrac{1}{2}$ ${}\Rightarrow m^2$ ${}= 2$
$m = \sqrt{2}$
F(Q₃) = ¾. $Q_3 = \sqrt{3}$

Part (c)1 mark

$1 - F(1) = \frac{3}{4}$

Total6

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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